Clarify the terminology of “cofinal”

[ayberkt]

As pointed out by @tomdjong, and discussed with @martinescardo.

[ayberkt]

@martinescardo @tomdjong: How would you like the term “is refined by” instead of “is cofinal in”, by analogy with the lattice of partitions?

One could also just switch to the opposite order and say “refines”.

[tomdjong]

I find "exceeds" (as originally suggested) to be a little bit clearer because "refines" only makes sense in the context of an information order. For example: consider the powerset of some set X, the family a : 1 -> P(X) that picks out the subset X, and the singleton family b : X -> P(X). Then a exceeds b, but it doesn't make so much sense to say that b is refined by a, right?

[ayberkt]

Okay, exceeds is also good. I’ll rename it to exceeds. I just wanted to put this suggestion out there.

Where I was coming from was the fact that this terminology is used in point-set topology (see the definition at the end of pg. 245 of Munkres's book on topology). So I wasn’t thinking about the information ordering at all.